\(\int x^4 (d+e x)^3 (d^2-e^2 x^2)^{5/2} \, dx\) [66]

Optimal result

Integrand size = 27, antiderivative size = 281 \[ \int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {27 d^{13} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^5} \]

[Out]

9/512*d^9*x*(-e^2*x^2+d^2)^(3/2)/e^4+9/640*d^7*x*(-e^2*x^2+d^2)^(5/2)/e^4-20/143*d^4*x^2*(-e^2*x^2+d^2)^(7/2)/
e^3-9/40*d^3*x^3*(-e^2*x^2+d^2)^(7/2)/e^2-45/143*d^2*x^4*(-e^2*x^2+d^2)^(7/2)/e-1/4*d*x^5*(-e^2*x^2+d^2)^(7/2)
-1/13*e*x^6*(-e^2*x^2+d^2)^(7/2)-1/320320*d^5*(27027*e*x+12800*d)*(-e^2*x^2+d^2)^(7/2)/e^5+27/1024*d^13*arctan
(e*x/(-e^2*x^2+d^2)^(1/2))/e^5+27/1024*d^11*x*(-e^2*x^2+d^2)^(1/2)/e^4

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1823, 847, 794, 201, 223, 209} \[ \int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {27 d^{13} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^5}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}+\frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2} \]

[In]

Int[x^4*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(27*d^11*x*Sqrt[d^2 - e^2*x^2])/(1024*e^4) + (9*d^9*x*(d^2 - e^2*x^2)^(3/2))/(512*e^4) + (9*d^7*x*(d^2 - e^2*x
^2)^(5/2))/(640*e^4) - (20*d^4*x^2*(d^2 - e^2*x^2)^(7/2))/(143*e^3) - (9*d^3*x^3*(d^2 - e^2*x^2)^(7/2))/(40*e^
2) - (45*d^2*x^4*(d^2 - e^2*x^2)^(7/2))/(143*e) - (d*x^5*(d^2 - e^2*x^2)^(7/2))/4 - (e*x^6*(d^2 - e^2*x^2)^(7/
2))/13 - (d^5*(12800*d + 27027*e*x)*(d^2 - e^2*x^2)^(7/2))/(320320*e^5) + (27*d^13*ArcTan[(e*x)/Sqrt[d^2 - e^2
*x^2]])/(1024*e^5)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 847

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^
m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^4 \left (d^2-e^2 x^2\right )^{5/2} \left (-13 d^3 e^2-45 d^2 e^3 x-39 d e^4 x^2\right ) \, dx}{13 e^2} \\ & = -\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^4 \left (351 d^3 e^4+540 d^2 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{156 e^4} \\ & = -\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^3 \left (-2160 d^4 e^5-3861 d^3 e^6 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{1716 e^6} \\ & = -\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^2 \left (11583 d^5 e^6+21600 d^4 e^7 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{17160 e^8} \\ & = -\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x \left (-43200 d^6 e^7-104247 d^5 e^8 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{154440 e^{10}} \\ & = -\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^7\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{320 e^4} \\ & = \frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (9 d^9\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{128 e^4} \\ & = \frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^{11}\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{512 e^4} \\ & = \frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^{13}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{1024 e^4} \\ & = \frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^{13}\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^4} \\ & = \frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {27 d^{13} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.68 \[ \int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-204800 d^{12}-135135 d^{11} e x-102400 d^{10} e^2 x^2-90090 d^9 e^3 x^3-76800 d^8 e^4 x^4+952952 d^7 e^5 x^5+2498560 d^6 e^6 x^6+816816 d^5 e^7 x^7-2938880 d^4 e^8 x^8-2690688 d^3 e^9 x^9+430080 d^2 e^{10} x^{10}+1281280 d e^{11} x^{11}+394240 e^{12} x^{12}\right )-270270 d^{13} \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{5125120 e^5} \]

[In]

Integrate[x^4*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-204800*d^12 - 135135*d^11*e*x - 102400*d^10*e^2*x^2 - 90090*d^9*e^3*x^3 - 76800*d^8*e^4
*x^4 + 952952*d^7*e^5*x^5 + 2498560*d^6*e^6*x^6 + 816816*d^5*e^7*x^7 - 2938880*d^4*e^8*x^8 - 2690688*d^3*e^9*x
^9 + 430080*d^2*e^10*x^10 + 1281280*d*e^11*x^11 + 394240*e^12*x^12) - 270270*d^13*ArcTan[(e*x)/(Sqrt[d^2] - Sq
rt[d^2 - e^2*x^2])])/(5125120*e^5)

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.66

method	result	size
risch	\(-\frac {\left (-394240 e^{12} x^{12}-1281280 d \,e^{11} x^{11}-430080 d^{2} e^{10} x^{10}+2690688 d^{3} e^{9} x^{9}+2938880 d^{4} e^{8} x^{8}-816816 d^{5} e^{7} x^{7}-2498560 d^{6} e^{6} x^{6}-952952 d^{7} e^{5} x^{5}+76800 d^{8} e^{4} x^{4}+90090 d^{9} e^{3} x^{3}+102400 d^{10} e^{2} x^{2}+135135 d^{11} e x +204800 d^{12}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5125120 e^{5}}+\frac {27 d^{13} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{1024 e^{4} \sqrt {e^{2}}}\)	\(185\)
default	\(e^{3} \left (-\frac {x^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{13 e^{2}}+\frac {6 d^{2} \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )}{11 e^{2}}\right )}{13 e^{2}}\right )+d^{3} \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )+3 d \,e^{2} \left (-\frac {x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )}{12 e^{2}}\right )+3 d^{2} e \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )}{11 e^{2}}\right )\)	\(548\)

[In]

int(x^4*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/5125120*(-394240*e^12*x^12-1281280*d*e^11*x^11-430080*d^2*e^10*x^10+2690688*d^3*e^9*x^9+2938880*d^4*e^8*x^8
-816816*d^5*e^7*x^7-2498560*d^6*e^6*x^6-952952*d^7*e^5*x^5+76800*d^8*e^4*x^4+90090*d^9*e^3*x^3+102400*d^10*e^2
*x^2+135135*d^11*e*x+204800*d^12)/e^5*(-e^2*x^2+d^2)^(1/2)+27/1024*d^13/e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(
-e^2*x^2+d^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.65 \[ \int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {270270 \, d^{13} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (394240 \, e^{12} x^{12} + 1281280 \, d e^{11} x^{11} + 430080 \, d^{2} e^{10} x^{10} - 2690688 \, d^{3} e^{9} x^{9} - 2938880 \, d^{4} e^{8} x^{8} + 816816 \, d^{5} e^{7} x^{7} + 2498560 \, d^{6} e^{6} x^{6} + 952952 \, d^{7} e^{5} x^{5} - 76800 \, d^{8} e^{4} x^{4} - 90090 \, d^{9} e^{3} x^{3} - 102400 \, d^{10} e^{2} x^{2} - 135135 \, d^{11} e x - 204800 \, d^{12}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5125120 \, e^{5}} \]

[In]

integrate(x^4*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/5125120*(270270*d^13*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (394240*e^12*x^12 + 1281280*d*e^11*x^11 +
430080*d^2*e^10*x^10 - 2690688*d^3*e^9*x^9 - 2938880*d^4*e^8*x^8 + 816816*d^5*e^7*x^7 + 2498560*d^6*e^6*x^6 +
952952*d^7*e^5*x^5 - 76800*d^8*e^4*x^4 - 90090*d^9*e^3*x^3 - 102400*d^10*e^2*x^2 - 135135*d^11*e*x - 204800*d^
12)*sqrt(-e^2*x^2 + d^2))/e^5

Sympy [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.02 \[ \int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\begin {cases} \frac {27 d^{13} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{1024 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {40 d^{12}}{1001 e^{5}} - \frac {27 d^{11} x}{1024 e^{4}} - \frac {20 d^{10} x^{2}}{1001 e^{3}} - \frac {9 d^{9} x^{3}}{512 e^{2}} - \frac {15 d^{8} x^{4}}{1001 e} + \frac {119 d^{7} x^{5}}{640} + \frac {488 d^{6} e x^{6}}{1001} + \frac {51 d^{5} e^{2} x^{7}}{320} - \frac {82 d^{4} e^{3} x^{8}}{143} - \frac {21 d^{3} e^{4} x^{9}}{40} + \frac {12 d^{2} e^{5} x^{10}}{143} + \frac {d e^{6} x^{11}}{4} + \frac {e^{7} x^{12}}{13}\right ) & \text {for}\: e^{2} \neq 0 \\\left (\frac {d^{3} x^{5}}{5} + \frac {d^{2} e x^{6}}{2} + \frac {3 d e^{2} x^{7}}{7} + \frac {e^{3} x^{8}}{8}\right ) \left (d^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]

[In]

integrate(x**4*(e*x+d)**3*(-e**2*x**2+d**2)**(5/2),x)

[Out]

Piecewise((27*d**13*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0))
, (x*log(x)/sqrt(-e**2*x**2), True))/(1024*e**4) + sqrt(d**2 - e**2*x**2)*(-40*d**12/(1001*e**5) - 27*d**11*x/
(1024*e**4) - 20*d**10*x**2/(1001*e**3) - 9*d**9*x**3/(512*e**2) - 15*d**8*x**4/(1001*e) + 119*d**7*x**5/640 +
488*d**6*e*x**6/1001 + 51*d**5*e**2*x**7/320 - 82*d**4*e**3*x**8/143 - 21*d**3*e**4*x**9/40 + 12*d**2*e**5*x*
*10/143 + d*e**6*x**11/4 + e**7*x**12/13), Ne(e**2, 0)), ((d**3*x**5/5 + d**2*e*x**6/2 + 3*d*e**2*x**7/7 + e**
3*x**8/8)*(d**2)**(5/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.91 \[ \int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {1}{13} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{6} - \frac {1}{4} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{5} + \frac {27 \, d^{13} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{1024 \, \sqrt {e^{2}} e^{4}} + \frac {27 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{11} x}{1024 \, e^{4}} - \frac {45 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{4}}{143 \, e} + \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{9} x}{512 \, e^{4}} - \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x^{3}}{40 \, e^{2}} + \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{7} x}{640 \, e^{4}} - \frac {20 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4} x^{2}}{143 \, e^{3}} - \frac {27 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{5} x}{320 \, e^{4}} - \frac {40 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{6}}{1001 \, e^{5}} \]

[In]

integrate(x^4*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

-1/13*(-e^2*x^2 + d^2)^(7/2)*e*x^6 - 1/4*(-e^2*x^2 + d^2)^(7/2)*d*x^5 + 27/1024*d^13*arcsin(e^2*x/(d*sqrt(e^2)
))/(sqrt(e^2)*e^4) + 27/1024*sqrt(-e^2*x^2 + d^2)*d^11*x/e^4 - 45/143*(-e^2*x^2 + d^2)^(7/2)*d^2*x^4/e + 9/512
*(-e^2*x^2 + d^2)^(3/2)*d^9*x/e^4 - 9/40*(-e^2*x^2 + d^2)^(7/2)*d^3*x^3/e^2 + 9/640*(-e^2*x^2 + d^2)^(5/2)*d^7
*x/e^4 - 20/143*(-e^2*x^2 + d^2)^(7/2)*d^4*x^2/e^3 - 27/320*(-e^2*x^2 + d^2)^(7/2)*d^5*x/e^4 - 40/1001*(-e^2*x
^2 + d^2)^(7/2)*d^6/e^5

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.63 \[ \int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {27 \, d^{13} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{1024 \, e^{4} {\left | e \right |}} - \frac {1}{5125120} \, {\left (\frac {204800 \, d^{12}}{e^{5}} + {\left (\frac {135135 \, d^{11}}{e^{4}} + 2 \, {\left (\frac {51200 \, d^{10}}{e^{3}} + {\left (\frac {45045 \, d^{9}}{e^{2}} + 4 \, {\left (\frac {9600 \, d^{8}}{e} - {\left (119119 \, d^{7} + 2 \, {\left (156160 \, d^{6} e + 7 \, {\left (7293 \, d^{5} e^{2} - 8 \, {\left (3280 \, d^{4} e^{3} + {\left (3003 \, d^{3} e^{4} - 10 \, {\left (48 \, d^{2} e^{5} + 11 \, {\left (4 \, e^{7} x + 13 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]

[In]

integrate(x^4*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

27/1024*d^13*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^4*abs(e)) - 1/5125120*(204800*d^12/e^5 + (135135*d^11/e^4 + 2*(512
00*d^10/e^3 + (45045*d^9/e^2 + 4*(9600*d^8/e - (119119*d^7 + 2*(156160*d^6*e + 7*(7293*d^5*e^2 - 8*(3280*d^4*e
^3 + (3003*d^3*e^4 - 10*(48*d^2*e^5 + 11*(4*e^7*x + 13*d*e^6)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*sqrt(-e^2*x^2 +
d^2)

Mupad [F(-1)]

Timed out. \[ \int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int x^4\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \]

[In]

int(x^4*(d^2 - e^2*x^2)^(5/2)*(d + e*x)^3,x)

[Out]

int(x^4*(d^2 - e^2*x^2)^(5/2)*(d + e*x)^3, x)